3.1855 $$\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^2} \, dx$$

Optimal. Leaf size=54 $\frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2}$

[Out]

((c*d^2 - a*e^2)*(a*e + c*d*x)^4)/(4*c^2*d^2) + (e*(a*e + c*d*x)^5)/(5*c^2*d^2)

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Rubi [A]  time = 0.0312005, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $\frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^2,x]

[Out]

((c*d^2 - a*e^2)*(a*e + c*d*x)^4)/(4*c^2*d^2) + (e*(a*e + c*d*x)^5)/(5*c^2*d^2)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^2} \, dx &=\int (a e+c d x)^3 (d+e x) \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{c d}+\frac{e (a e+c d x)^4}{c d}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac{e (a e+c d x)^5}{5 c^2 d^2}\\ \end{align*}

Mathematica [A]  time = 0.0259652, size = 79, normalized size = 1.46 $\frac{1}{20} x \left (10 a^2 c d e^2 x (3 d+2 e x)+10 a^3 e^3 (2 d+e x)+5 a c^2 d^2 e x^2 (4 d+3 e x)+c^3 d^3 x^3 (5 d+4 e x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^2,x]

[Out]

(x*(10*a^3*e^3*(2*d + e*x) + 10*a^2*c*d*e^2*x*(3*d + 2*e*x) + 5*a*c^2*d^2*e*x^2*(4*d + 3*e*x) + c^3*d^3*x^3*(5
*d + 4*e*x)))/20

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Maple [B]  time = 0.04, size = 136, normalized size = 2.5 \begin{align*}{\frac{{c}^{3}{d}^{3}e{x}^{5}}{5}}+{\frac{ \left ( 2\,a{c}^{2}{d}^{2}{e}^{2}+{c}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}{e}^{3}dc+2\,acde \left ( a{e}^{2}+c{d}^{2} \right ) +{c}^{2}{d}^{3}ae \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +2\,{a}^{2}c{d}^{2}{e}^{2} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^2,x)

[Out]

1/5*c^3*d^3*e*x^5+1/4*(2*a*c^2*d^2*e^2+c^2*d^2*(a*e^2+c*d^2))*x^4+1/3*(a^2*e^3*d*c+2*a*c*d*e*(a*e^2+c*d^2)+c^2
*d^3*a*e)*x^3+1/2*(a^2*e^2*(a*e^2+c*d^2)+2*a^2*c*d^2*e^2)*x^2+a^3*e^3*d*x

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Maxima [A]  time = 1.08454, size = 128, normalized size = 2.37 \begin{align*} \frac{1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac{1}{4} \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} +{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^2,x, algorithm="maxima")

[Out]

1/5*c^3*d^3*e*x^5 + a^3*d*e^3*x + 1/4*(c^3*d^4 + 3*a*c^2*d^2*e^2)*x^4 + (a*c^2*d^3*e + a^2*c*d*e^3)*x^3 + 1/2*
(3*a^2*c*d^2*e^2 + a^3*e^4)*x^2

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Fricas [A]  time = 1.37772, size = 193, normalized size = 3.57 \begin{align*} \frac{1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac{1}{4} \,{\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} +{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/5*c^3*d^3*e*x^5 + a^3*d*e^3*x + 1/4*(c^3*d^4 + 3*a*c^2*d^2*e^2)*x^4 + (a*c^2*d^3*e + a^2*c*d*e^3)*x^3 + 1/2*
(3*a^2*c*d^2*e^2 + a^3*e^4)*x^2

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Sympy [B]  time = 0.144167, size = 100, normalized size = 1.85 \begin{align*} a^{3} d e^{3} x + \frac{c^{3} d^{3} e x^{5}}{5} + x^{4} \left (\frac{3 a c^{2} d^{2} e^{2}}{4} + \frac{c^{3} d^{4}}{4}\right ) + x^{3} \left (a^{2} c d e^{3} + a c^{2} d^{3} e\right ) + x^{2} \left (\frac{a^{3} e^{4}}{2} + \frac{3 a^{2} c d^{2} e^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**2,x)

[Out]

a**3*d*e**3*x + c**3*d**3*e*x**5/5 + x**4*(3*a*c**2*d**2*e**2/4 + c**3*d**4/4) + x**3*(a**2*c*d*e**3 + a*c**2*
d**3*e) + x**2*(a**3*e**4/2 + 3*a**2*c*d**2*e**2/2)

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Giac [B]  time = 1.19977, size = 197, normalized size = 3.65 \begin{align*} \frac{1}{20} \,{\left (4 \, c^{3} d^{3} - \frac{15 \,{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{5} e^{\left (-4\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^2,x, algorithm="giac")

[Out]

1/20*(4*c^3*d^3 - 15*(c^3*d^4*e - a*c^2*d^2*e^3)*e^(-1)/(x*e + d) + 20*(c^3*d^5*e^2 - 2*a*c^2*d^3*e^4 + a^2*c*
d*e^6)*e^(-2)/(x*e + d)^2 - 10*(c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*e^(-3)/(x*e + d)^3)
*(x*e + d)^5*e^(-4)